We propose a novel hematoxylin and eosin (H&E) stain normalization method based on a modified U-Net neural network architecture. Unlike previous deep-learning methods that were often based on generative adversarial networks (GANs), we take a teacher-student approach and use paired datasets generated by a trained CycleGAN to train a U-Net to perform the stain normalization task. Through experiments, we compared our method to two recent competing methods, CycleGAN and StainNet, a lightweight approach also based on the teacher-student model. We found that our method is faster and can process larger images with better quality compared to CycleGAN. We also compared to StainNet and found that our method delivered quantitatively and qualitatively better results.